1. Math 231: Differential Equations Set 1: Basic Ideas Notes abridged from the Power Point Notes of Dr. Richard Rubin

2. 1st Order Differential Equations We start with a definition: A differential equation is an equation which contains derivatives. Look at the examples Table 1 pg 2 of the Supplement. Do any look familiar?

3. Order Differential Equations A differential equation is an equation which contains derivatives. Table 1 pg 2 contains a number of examples of differential equations. Note the 1st equation in Table 1. Where have we seen it before?

5. 1st Order Differential Equations Another example from nuclear engineering (Radioactive Decay): In this equation, we know N and t are variables.  is a constant which changes for different materials.

6. 1st Order Differential Equations Another example from nuclear engineering (Radioactive Decay): We have a special name for numbers like  Parameters are quantities which don’t vary for a particular problem but which may vary from problem to problem.

7. Differential Equations Quantities which don’t vary for a particular problem but which may vary from problem to problem are called parameters. Another example from mechanics: State which quantities are independent and dependent variables and which are parameters.

8. Order of Differential Equations An example from nuclear engineering: The order of differential equation is determined by the highest derivative in the equation. The above equation is said to be 2nd order.

9. Ordinary Derivative Definition: An ordinary derivative is a derivative of a function of just one independent variable with respect to that variable. Examples:

10. Definition: An ordinary derivative is a derivative of a function of just one independent variable with respect to that variable. Examples (not ordinary derivatives- paratial derivatives):

11. Definition: An ordinary differential equation is an equation that contains ordinary derivatives of one or more dependent variables all of which are with respect to the same independent variable. DE with dependent variables which depend on more than one independent variable are not ordinary DE!

12. Example of an Ordinary DE: Relatavistic differential equation for the speed v of a rocket in linear motion: m= mass of fuel burned, v=v(m), v e = exhaust speed, and c = speed of light.

13. Example of a Non- ordinary DE: One dimensional equation for wave displacement, u(x,t): a determines the constant velocity of the wave.

14. 1st Order Differential Equations Instantaneous rates of change are associated with a derivative with respect to the independent variable. Models of real phenomena often contain rates of change. Examples of such things are: Temperature Change (text pp. 4-6)

15. 1st Order Differential Equations We will start our consideration of how to solve ordinary differential equations by considering the simplest possible type: How many solutions are there?

16. 1st Order Differential Equations Now realize that you already know quite a lot about solving differential equations! All of the integration techniques you learned in calculus are applicable to solving differential equations of this type!

17. 1st Order Differential Equations We will start our consideration of how to solve ordinary differential equations by considering the simplest possible type: Do you remember: Method of Substitution? Example:

18. 1st Order Differential Equations We will start our consideration of how to solve ordinary differential equations by considering the simplest possible type: Do you remember: Method of Partial Fractions? Example:

19. 1st Order Differential Equations We will start our consideration of how to solve ordinary differential equations by considering the simplest possible type: Do you remember: Integration by Parts? Example: